An Algebraic Proof of Weierstrass's Approximation Theorem

TL;DR

This paper presents a novel algebraic proof of Weierstrass's approximation theorem using Vandermonde matrices, demonstrating polynomial approximation of continuous functions and their relation to Taylor series.

Contribution

The paper introduces an algebraic approach leveraging Vandermonde matrices to prove Weierstrass's theorem, offering new insights into polynomial approximation methods.

Findings

Vandermonde matrices have non-zero determinants under specific conditions.

Constructed polynomial sequences algebraically approach the Taylor series.

Method applies to infinitely differentiable functions, confirming approximation accuracy.

Abstract

In this paper we use the Vandermonde matrices and their properties to give a new proof of the classical result of Karl Weierstrass about the approximation of continuous functions $f$ on closed intervals, using a sequence of polynomials. The proof solves linear systems of equations using that the Vandermonde matrices have always non zero determinants, when the entries of the power series of the rows of the matrix are all different. We provide several examples, and we also use our method to observe that the sequence of polynomials that we construct algebraically approaches the Taylor series of a function $f$ which is infinitely differentiable.